Question

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .

Answer 1

Stick length = 1 unit

Let the stick break at the distance x.

Since expected length of the shorter stick is needed x ϵ [0, 0.5]

For uniform distribution,

Probability density function = p(x) = \(\frac{1}{{b - a}} = \frac{1}{{0.5 - 0}} = 2\)

Expected length = E(x)

\(E\left( x \right) = \mathop \smallint \limits_0^{0.5} xp\left( x \right)dx\)

\(E\left( x \right) = \mathop \smallint \limits_0^{0.5} x2dx = 2{\left( {\frac{{{x^2}}}{2}} \right)_0}^{0.5} = 0.25\)

the expected length of the shorter stick is 0.25

Important Point:

Official GATE CS answer range is 0.24 to 0.27